On difference-differential Lax pairs and integrals of Painlevé equations in finite characteristic

2026/07/08 by Nalini Joshi, Tomas Lasic Latimer, Pieter Roffelsen · 1 voice
Physics and Astronomy · Mathematics · #nlin.SI #math-ph #math.DS

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Abstract

We collect rank two difference-differential Lax pairs for classical Painlevé equations in the literature and put each in 2× 2 matrix form with the coefficient matrix of the spectral equation a degree two matrix polynomial. We describe and apply a general method to obtain integrals of motion in characteristic p from these Lax pairs. For every relevant Painlevé equation, this leads to a countable list of integrals of motion, with one entry for each prime p.

Citations

• On integrals of non-autonomous dynamical systems in finite characteristic
• Arithmetic dynamics of a discrete Painlevé equation
• On the crystal limit of the q-difference sixth Painlevé equation
• Reduction of lattice equations to the Painlevé equations: P_{\rm IV} and P_{\rm V}
• A symmetric difference-differential Lax pair for Painlevé VI
• Integrability of Discrete Equations Modulo a Prime
• Proofs of power sum and binomial coefficient congruences via Pascal's identity
• Proofs of Power Sum and Binomial Coefficient Congruences Via Pascal's Identity
• Algebraic solutions of the sixth Painleve equation
• Algebraic solutions of the sixth Painlevé equation
• Moduli spaces of d-connections and difference Painleve equations
• The fifty-two icosahedral solutions to Painleve VI

Discussions

• Nalini Joshi, Tomas Lasic Latimer, Pieter Roffelsen: On difference-differential Lax pairs and integrals of Painlev\'e equations in finite characteristic https://arxiv.org/abs/2607.06980 https://arxiv. [bsky, 0 points, 0 comments]

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